Geometric and computational aspects of chiral topological quantum matter

Abstract

In this thesis, we study chiral topological phases of 2+1 dimensional quantum matter. Such phases are abstractly characterized by their non-vanishing chiral central charge $c$, a topological invariant which appears as the coefficient of a gravitational Chern-Simons (gCS) action in bulk, and of corresponding gravitational anomalies at boundaries. The chiral central charge is of particular importance in chiral superfluids and superconductors (CSF/Cs), where $U(1)$ particle-number symmetry is broken, and $c$ is, in some cases, the only topological invariant characterizing the system. However, as opposed to invariants which can be probed by gauge fields in place of gravity, the concrete physical implications of $c$ in the context of condensed matter physics is quite subtle, and has been the subject of ongoing research and controversy. The first two parts of this thesis are devoted to the physical interpretation of the gCS action and gravitational anomalies in the context of CSF/Cs, where they are of particular importance, but have nevertheless remained poorly understood. We then turn to a seemingly unrelated aspect of chiral topological phases - their computational complexity. The infamous $signproblem$ leads to an exponential complexity in Monte Carlo simulations of generic many-body quantum systems. Nevertheless, many phases of matter are known to admit a sign-problem-free representative, allowing an efficient classical simulation. The possibility of $intrinsic$ sign problems, where a phase of matter admits no sign-problem-free representative, was recently raised but remains largely unexplored. Here, we establish the existence of an intrinsic sign problem in a broad class of chiral topological phases, both bosonic and fermionic, defined by the requirement that $e^{2\pi ic/24}$ is $not$ the topological spin of an anyon.

Thesis. 88 pages, 11 figures, 1 table (+ appendices: 64 pages, 2 figures)

References in corpus (42)

• Non-Abelian Anyons and Topological Quantum Computation
• Topological Field Theory of Time-Reversal Invariant Insulators
• Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations
• Non-Abelian Topological Order in S-Wave Superfluids of Ultracold Fermionic Atoms
• Three-Dimensional Gravity Revisited
• $p_x+i{p}_y$ superfluid from s-wave interactions of fermionic cold atoms
• General coordinate invariance and conformal invariance in nonrelativistic physics: Unitary Fermi gas
• Quantum limit of heat flow across a single electronic channel
• Characterizing topological order by studying the ground states of an infinite cylinder
• xPert: Computer algebra for metric perturbation theory
• Spacetime Symmetries of the Quantum Hall Effect
• Framing Anomaly in the Effective Theory of Fractional Quantum Hall Effect
• Thermal Hall Effect and Geometry with Torsion
• Low-energy effective theory in the bulk for transport in a topological phase
• Bridging lattice-scale physics and continuum field theory with quantum Monte Carlo simulations
• Topological field theory and thermal responses of interacting topological superconductors
• Collective modes and electromagnetic response of a chiral superconductor
• Transport signatures of Hall viscosity
• Topological central charge from Berry curvature: gravitational anomalies in trial wavefunctions for topological phases
• Torsion, Parity-odd Response and Anomalies in Topological States
• Hall viscosity, topological states and effective theories
• Orbital Angular Momentum and Spectral Flow in Two Dimensional Chiral Superfluids
• Melting of interference in the fractional quantum Hall effect: Appearance of neutral modes
• Gauge-invariant electromagnetic response of a chiral px+ipy superconductor
• FQHE on curved backgrounds, free fields and large N
• Guiding-center Hall viscosity and intrinsic dipole moment along edges of incompressible fractional quantum Hall fluids
• Symmetries and boundary theories for chiral Projected Entangled Pair States
• Phenomenology of non-relativistic parity-violating hydrodynamics in 2+1 dimensions
• "Hall viscosity" and intrinsic metric of incompressible fractional Hall fluids
• Hall viscosity and electromagnetic response of electrons in graphene
• Effective theory of two-dimensional chiral superfluids: gauge duality and Newton-Cartan formulation
• Protection of topological order by symmetry and many-body localization
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• On Nambu's Fermion-Boson Relations for Superfluid ${}^3$He-B
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• More On Nonrelativistic Diffeomorphism Invariance
• Hall Viscosity and the Acoustic Faraday Effect
• The Cosmological Constant and Lorentz Invariance of the Vacuum State
• Hall viscosity, spin density, and torsion
• Orbital angular momentum in a nonchiral topological superconductor
• The free surface of a colloidal chiral fluid: waves and instabilities from odd stress and Hall viscosity
• Equivalence between vortices, twists and chiral gauge fields in Kitaev's honeycomb lattice model